Are there any special instances where a function will have more than two horizontal asymptotes?
a single horizontal asymptote occurs when $\displaystyle \lim_{x \to \pm \infty} f(x) = L$
it is possible for two to exist ...
$\displaystyle \lim_{x \to \infty} f(x) = L$
$\displaystyle \lim_{x \to -\infty} f(x) = K$
$\displaystyle K \ne L$
if there were three, then neither of the above limits would exist.
Do you know the vertical line test for a function?
What would more than two horizontal asymptotes imply?
Have a look at this.
Your question has nothing to do with "multivariable or vector Calculus", it has only to do with the definition of "horizontal asymptote".
If you want to ask your question in terms of multiple independent variables, you will have to define "horizontal" an "horizontal asymptote" in multiple variables.